Transcript Heuristic Optimisation in Design and Analysis

Quantum Key Distribution (QKD)
John A Clark
Dept. of Computer Science
University of York, UK
[email protected]
Communication
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The only really secure cryptosystem is the one-time pad
(provided you use it only once, which hasn’t always been the
case).
Essentially both participants possess the same random bit
stream b1 b2 b3 b4…..
The sender has a message m1 m2 m3 m4 ….
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Encodes message as c1 c2 c3 c4
c1  (b1 m1),c2  (b2  m2),c3  (b3  m3),c4  (b4  m4)
Receiver applies b1 b2 b3 b4 to obtain message
m1  (b1 c1),m2  (b2  c2),m3  (b3  c3),m4  (b4  c4)
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But how can we distribute this keystream b1 b2 b3 b4…?
When Alice met Bob
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Communicants will (following tradition) be Alice and Bob, trying
to communicate their love…
Alice
Bob
Eve
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Eve isn’t happy about this. She wants to listen in and interfere
Basic Scheme
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Basic scheme based on polarisation of photons
Photons are transverse magnetic waves – magnetic
and electric fields are perpendicular to the direction of
propagation. Also they are perpendicular to each
other.
y
x
z
Photons
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We will assume that we are dealing with linearly polarised light
but other schemes are possible (e.g. with circularly polarised
light).
We need to create photons that with an electric field oscillating
in the desired magnetic plane.
One way to do this is by passing light through an appropriate
polariser
Only vertically
polarised photons
emerge
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More sophisticated way is to use a Pockels Cell.
Detecting Photons
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Possible to detect absorption by using a Calcite crystal
Photon Detector
Photon Detector
Measuring a Photon
Suppose photon has polarisation
at angle q to a horizontal filter.
q
Measured as a 0 (absorbed)
with prob=sin2 q.
Measured as a 1 (permitted)
with prob=cos2 q.
Blocking is Freedom
Intensity
1.0
Intensity
0.5
Intensity
0
Intensity
0.125
Basic Scheme
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Basic scheme assumes that the polarisation of
photons can be arranged. For example
Vertical Polarisation
denotes 0
Horizontal Polarisation
denotes 1
Rectilinear Basis
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Suppose now that Alice sends a 0 in this scheme and
that Bob uses a photon detector with the same basis.
Alice
Sends
0
Bob
Receives
0
Alice
Sends
1
Bob
Receives
1
Diagonal Basis
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Can also arrange this with a diagonal basis
Alice
Sends
0
Bob
Receives
0
Alice
Sends
1
Bob
Receives
1
Basis Mismatch
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What if Alice and Bob choose different bases?
Alice Sends
0
Bob
Receives
0
Bob
Receives
1
Each result with probability 1/2
Use of Basis Summary
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A sender can encode a 0 or a 1 by choosing the
polarisation of the photon with respect to a basis
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The receiver Bob can observe (measure) the
polarisation with respect to either basis.
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Vertical => 0 Horizontal => 1; or
45 degrees => 0, 135o =>1
If same basis then bits are correctly received
If different basis then only 50% of bits are correctly
received.
This notion underpins one of the basic quantum
cryptography key distribution schemes.
What’s Eve up To?
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Now Eve gets in on the act and chooses to measure
the photon against some basis and then retransmit to
Bob.
Eve’s Dropping In
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Alice
Sends
0
Alice
Sends
1
Suppose Eve listens in using the same basis as Alice,
measures the photon and retransmits a photon as
measured (she goes undetected)
Eve
Measures
0
Eve
Measures
1
To Bob
To Bob
Eve’s Dropping In
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Suppose Eve listens in using a different basis to Alice
Alice Sends
0
Eve Measures
0
To Bob
Eve Measures
1
To Bob
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0 and 1
equally
likely
results
0 and 1
equally
likely
results
Similarly if Alice sends a 1 (or if Alice uses diagonal basis and
Eve uses rectilinear one)
Summary of Eve’s Droppings
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If Eve gets the basis wrong, then even if Bob gets
the same basis as Alice his measurements will only
be 50 percent correct.
If Alice and Bob become aware of such a mismatch
they will deduce that Eve is at work.
A scheme can be created to exploit this.
Alice and Bob
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To send and receive a photon Alice and Bob choose a
basis randomly. Alice sends a 0 or 1 using her basis
and Bob uses his basis to measure it.
Alice records the basis she used and the value sent.
Bob records the basis he used and the value he
measured.
When We are in Harmony
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Throw away results when bases disagree and keep
results when bases agree
Keep Value
Discard Value
Discard Value
Keep Value
Alice
Bob
We Agree
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Alice and Bob exchange a sequence of bit values
encoded in photon polarisation with bases chosen at
random.
Bob announces via an unjammable channel which
bases he used in each case.
Alice tells Bob whether choices of basis were correct.
They throw away any bit values where the basis
choice disagreed and keep those bit values were the
basis choice agreed.
Has Eve Listened In?
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Now we need to determine whether Eve has been
listening in.
How might this be done?
Has Eve Listened In?
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Can pick some bits at random and tell each other
what values were sent and received.
Sufficiently many mismatches then high chance of
Eve at work.
Has Eve Listened In?
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Can pick some random subset and determine the
parity of the bit values sent and received.
If parities disagree then Eve may have been at work
or else there has been an error.
Even if agree, parity information has been publicly
broadcast – so we discard the final contributing bit.
Can repeat this process numerous times to gain
increased confidence.
Creating Photons
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In practice creating a single photon may not be that
easy.
Can be done with dim light pulses.
But if two photons get created one can be captured
and measured whilst the other goes through to Alice.
They would both have the same polarisation so the
security here would be broken.
Keeping it All in Line
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The kit used to carry out key distribution way may be
rather sensitive to disturbance.
May need continuous adjustment to maintain right
physical set up etc.
Entangled States
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We have described the best known of protocols for
key distribution.
Various others are possible. For example, based on
entanglement with elements of an entangled pair
sent to each of Bob and Alice.
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Scheme due to Artur Ekert (Oxford).
General Usage
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Significant interest in QKD.
We don’t need to use it for everything.
Can use it to distribute key distribution keys.
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Keys we can use to carry out conventional key distribution
protocols securely.
Note: no prior contact is necessary.
Aside
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QKD here relies on being able to detect Eve’s
interfering.
Possible to go to other extreme and assume that data
will be intercepted:
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More conventional schemes proposed where trillions of bits
per second would be transmitted and only sender and
receiver know the (very small) time window for the key.
Idea is to swamp an interceptor with so much data that they
cannot possibly cope.
Summary
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Have outlined basics of a photon-based scheme that
allows a key to be created and shared between two
communicants in a manner that allows
eavesdropping to be detected.
Makes use of one of the fundamental features of
quantum mechanics
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Looking (measuring) disturbs things
QKD works!
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Experiments over 10’s of kilometres using fibre optics.
Work also in free space. Aim for QKD with low orbiting
satellites.